How do I use this to prove that Q[sqrt(2)] is not isomorphic to Q[sqrt(3)]? What property of a ring fails?

Thanks for the response from before.

This is an interesting problem that and are isomorphic as vector spaces but not as fields. The way you show they are not isomorphic is by assuming that is an isomorphism. First . Therefore, for . It is easy to show for by considering the negatives. Next . Thus, is an identity map. Now . In both cases we never get an onto mapping because nothing gets mapped to . A contradiction.